Computational Geometry (236719)
Prof. Gill Barequet (
Maor Grinberg (
Fall 2011-12


Fundamental techniques, data structures, and algorithms for solving geometric problems such as computing convex hulls, intersection of line segments, the Voronoi diagram and Delaunay triangulation of a point set, polygon triangulation, range search, linear programming, and point location. Some topics of discrete geometry, e.g., the crossing number of a graph and its applications, are also covered.

News and Messages

(9/Jan/12) Next semester I will give an advanced seminar (236801) on polyominoes and polycubes - see syllabus here. All students are encouraged to attend this seminar; note that the registration is manual.

(4/Jan/12) Due to a popular demand, all students may submit Ex4 till January 15, 12:00, without penalty in grade.

(1/Jan/12) A clarification about the Delaunay triangulation of four cocircular points can be found here.

(27/Dec/11) Ex3 was posted in the web page of the course. Enjoy!

(27/Dec/11) A recitation will be held on Thursday, January 19, 2012, and will be dedicated to solving exam questions.

(15/Dec/11) The lecture of Thursday 12/Jan/2012 is canceled.

(15/Dec/11) Maor began to maintain a FAQ file for ex4. See link below.

(7/Dec/11) Ex2 was posted in the web page of the course. Enjoy!

(2/Dec/11) Ex4 was posted in the web page of the course. Enjoy!

(14/Nov/11) Tentative recitation dates: 17/Nov/11, 8/Dec/11, 22/Dec/11. More dates will be published later.

(8/Nov/11) Ex1 was posted in the web page of the course. Enjoy!

(6/Nov/11) The recitation moves permanently to Taub 8.

(6/Nov/11) The lecture of Thursday 17/Nov/2011 is canceled. A compensation lecture will be given on Wednesday 16/Nov/2011 12:30-14:30 in Taub 4.

(3/Nov/11) The strike of the junior staff is over! A recitation will be held today (immediately after the lecture).

(3/Nov/11) Reminder: students (and free listeners) are asked to join the mailing list of the course. (Tips about the exam will be distributed only via email.) See instructions below.

(21/Oct/11) All students (including free listeners) are kindly asked to join the mailing list of the course. For this to happen, please e-mail me your full name, id #, faculty, and degree toward which you study. (This is in addition to the formal registration to the course! One can leave this mailing list at any time.)


Main text book: Computational Geometry: Algorithms and Applications (3rd ed.), M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Springer-Verlag, 2008.
For background: Computational Geometry in C (2nd ed.), J. O'Rourke, Cambridge University Press, 2000.

Library links

Grading Policy

3 Home assignments (dry): ~12.5% (Takef, submission in singletons!!);
Running project (wet): ~12.5% (same);
Final exam: 75% (Moed A: Monday 06/Feb/12, 13:00, location TBA; Moed B: Thursday 15/Mar/12, time TBA, location TBA).

Assignment 1 (dry): given 10/Nov/11, due 24/Nov/11.

Assignment 2 (dry): given 08/Dec/11, due 29/Dec/11.

Assignment 3 (dry): given 29/Dec/11, due 19/Jan/12.

Assignment 4 (wet): given 08/Dec/11, due 08/Jan/12 (Graphics files, FAQ file)

Course summary and slides

(1) Introduction
What is Computational Geometry? Example problems and motivations. Naive, incremental, and divide-and-conquer convex-hull algorithms.

(2) Vectors, Plane sweep
Vector cross product and orientation test. Segment-intersection test. Convex-polygon queries. Plane-sweep paradigm. Segment-intersection algorithm.

(a) Planar graphs
Graph definition, planar graphs, Euler's formula, the DCEL structure.

(3) Polygon triangulation
The art-gallery theorem. Partitioning a simple polygon into monotone pieces. Triangulating a monotone polygon.

(4) Linear programming
What is linear programming. A D&C algorithm for half-planes intersection. An incremental algorithm for half-planes intersection. Randomized linear programming. Unbounded linear programming. Smallest enclosing disk of a 2D point set.

(b) Polygonal skeletons
Straight skeleton of a polygon and a polyhedron. Their complexities, and algorithms to compute them.

(5) Orthogonal range searching
1D range searching. 2D kd-trees. 2D Range trees.

(6) Point location
Slabs structure. Trapezoidal map. A randomized incremental algorithm for computing a trapezoidal map. Worst- and average-case Time/Space analysis of the algorithm. Handling degeneracies.

(7) Voronoi diagram
Definition and variants. A plane-sweep algorithm for computing the Voronoi diagram of a point set.

(c) Voronoi diagram
More and alternative definitions. Lloyd's algorithm.

(8) Duality
A point-line duality in the plane and its properties.

(9) Line Arrangements
Line arrangements and their properties. The zone theorem. Computing an arrangement of lines. Levels in line arrangements. Halfspace discrepancy and its dual problem.

(c) Space Partitioning
BSP tress, quadtress.

(10) Delaunay triangulation
Triangulation of a point set. Angle vector and the triangulation that maximizes it. Delaunay triangulation and its relation to the angle vector. A randomized incremental algorithm for computing the Delaunay triangulation.

(11) The crossing-number lemma
The crossing-number lemma and a few applications of it.

(12) 2-point site Voronoi diagrams
Some 2-point site distance functions and their respective Voronoi diagrams.

(13) A few theorems
The upper-bound theorem. Interpretations of Voronoi diagrams. Zone theorems. Envelopes of lines and planes.